Mathematical Modeling and Stability Analysis of a SIRV Epidemic Model with Non-linear Force of Infection and Treatment
Keywords:Vaccination, Local stability, Reproduction number, Steady states, Global stability
AbstractThis paper considers the Susceptible-Infected-Vaccinated-Recovered (SIRV) deterministic model with a non linear force of infection and treatment, where individual humans that are vaccinated losses their vaccination after some time and become vulnerable to infections. The basic reproduction number \(R_0\) obtained from the model system is an epidemic threshold that determines if a disease will continue to ravage the human population or not.\ The model state equations considered in this paper possess two steady-state solutions such that if \(R_0<1\), the infection-absent steady-state solutions are locally and globally asymptotically stable. Also, if \(R_0>1\), a unique infection-persistent steady-state solutions are established, which is also locally and globally asymptotically stable. Thus, it leads to the persistence of infections in the human host population. Finally, numerical simulations were carried out to validate our theoretical results.
E. Beretta and V. Cappasso, On the general structure of epidemic system: Global stability, Computers and mathematics with Applications 12 (1986), 677 – 694, DOI: 10.1016/0898-1221(86)90054-4.
B. Buonomo and S. Rionero, On the stability for SIRS epidemic models with general non linear incidence rate, Applied Mathematics and Computations 217 (2010), 4010 – 4016, DOI: 10.1016/j.amc.2010.10.007.
J. P. R. S. Rao and M. N. Kumar, A dynamic model for infectious diseases: The role of vaccination and treatment, Chaos, Solitons & Fractals 75 (2015), 34 – 49, DOI: 10.1016/j.chaos.2015.02.004.
C. Castillo, Z. Feng and W. Huang, On the computation of R0 and its role on global stability, http://www.math.la.asu.edu/chavez/2002/j3276.pdf (2003).
W. Derrick and S. I. Grossman, Elementary Differential Equations with Applications: Short Course, Addison-Wesley Publishing Company, Philippines (1976).
O. Diekmann, J. Hesterbeek and J. Metz, On the definition and computation of the basic reproduction number R0 in models for infectious disease, J. Math. Biol. 28 (1990), 365 – 382, DOI: 10.1007/BF00178324.
L. Esteva-Peralta and J. X. Velasco-Hernandez, M-Matrices and local stability in epidemic model, Math. Comp. Model. 36 (2002), 491 – 501, DOI: 10.1016/S0895-7177(02)00178-4.
H. W. Hethcote, The mathematics of infectious disease, SIAM Rev. 42 (2000), 599, DOI: 10.1137/S0036144500371907.
Z. Hu, W. Ma and S. Ruan, Analysis of SIR epidemic models with non linear incidence rate and treatment, Mathematical Biosciences 238 (2012), 12 – 22, DOI: 10.1016/j.mbs.2012.03.010.
A. Korobeinikov and D. K. Mani, Non linear incidence and stability of infectious disease models, Math. Med. Biol. 22 (2005), 113 – 128, DOI: 10.1093/imammb/dqi001.
Ranjith Kumar G., Lakshmi Narayan K. and Ravindra Reddy B., Dynamics of an SIS epidemic model with a saturated incidence rate under time delay and stochastic influence, International Journal of Pure and Applied Mathematics 112 (2017), 695 – 703, DOI: 10.12732/ijpam.v112i4.4.
J. P. La-Salle and S. Lefschetz, Stability by Liapunovs Direct Method, Academic press, New York (1961).
G.-H. Li and Y.-X. Zhang, Dynamic behavior of a modified SIR model in epidemic diseases using non linear incidence rate and treatment, PlosOne 12(4) (2017), e0175789, DOI: 10.1371/journal.pone.0175789.
Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific Publishing Co. Pte. Ltd. (2009), pages 512, DOI: 10.1142/6799.
P. Haukkanen and T. Tossavainen, A generalization of descartes rule of signs and fundamental theorem of algebra, Applied Mathematics and Computation 218 (2011), 1203 – 1207, DOI: 10.1016/j.amc.2011.05.107.
D. B. Prates, C. L. T. F. Jardim, L. A. F. Ferreira, J. M. da Silva and M. V. Kritz, Vaccination strategies: A comparative study in an epidemic scenario, J. Phys.: Conf. Ser. 738 (2016), 012 – 083, DOI: 10.1088/1742-6596/738/1/012083.
Mordern Automatic Control, Routh stability criterion, ECE680, (June 13, 2007), http://www.engr.iupui.edu/skoskie/ECE680/Routh.pdf.
P. Van den Driessche and J. Watmough, Reproduction Number and sub threshold epidemic equilibrium for compartmental models for disease transmission, Mathematical Biosciences 180 (2002), 29 – 48, DOI: 10.1016/S0025-5564(02)00108-6.
W. Yang, C. Sun and J. Arino, Global analysis for a general epidemiological model with vaccination and varying population, J. Math. Anal. Appl. 372 (2010), 208 – 223, DOI: 10.1016/j.jmaa.2010.07.017.
How to Cite
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.